The influence of inclinations on the dynamical stability of multi-planet systems

Abstract

ABSTRACT A type of compactly spaced and comparably sized multi-exoplanet system similar to TRAPPIST-1 has been discovered recently. The stability of these systems is an important issue, requiring further study. We examined how the initial inclinations influence the stability of multi-planet systems and derived an empirical formula describing the dependence of the instability time-scale on planetary mass, space separation and initial inclination. We find the following. (i) If space separations between planets are greater than 12RH (mutual Hill radius), coplanar multi-planet systems with 10−6 ≤ μ ≤ 10−3 (reduced planetary mass μ = m/M*) will remain stable within 1010Tin (the innermost orbital period). (ii) If initial inclinations of planets are smaller than 10° and space separations are greater than 10RH, multi-planet systems consisting of ≥5 planets with μ ≥ 10−5 will remain stable within 1010Tin. (iii) Initial inclinations in [0°, 10°] have inconsequential effects on the instability time-scales of massive multi-planet systems (μ ≥ 10−5), because eccentricities (excited during evolution) dominate the stability of these systems. (iv) If the initial inclinations are large enough (≥10°), sharp increases of instability time-scales in groups with 10−3 ≥ μ ≥ 10−5 will be moderated. This article presents a comprehensive study of the influence of inclination on the stability of multi-planet systems and discusses critical space separations for a multi-planet system becoming unstable.